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Message: 850 - Contents - Hide Contents Date: Fri, 24 Aug 2001 04:41:34 Subject: Re: Now I think "the hypothesis" is true :) From: Paul Erlich --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:> --- In tuning-math@y..., genewardsmith@j... wrote:>> --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote: >>>>> What makes the apparent relationship between MOS and >>> tempered periodicity blocks hard to fathom ... >> It's not so hard as I first thought. In MOS we are dealing with the > log of the generator over the log of the period. In PB's we are > dealing with vectors of logs (except they are all logs to different > prime bases).They can be anything. The just intervals forming the basis for the lattice are irrelevant for the hypothesis. They can be any intervals you want, as long as they're linearly independent.
Message: 851 - Contents - Hide Contents Date: Fri, 24 Aug 2001 04:49:24 Subject: Re: Jacks From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> Let me start by reviewing Farey sequences. The first row of the Farey > sequence is [0/1, 1/1] and the nth row is obtained by inserting the > fraction (p1+q1)/(p2+q2) between p1/q1 and p2/q2 if it is in reduced > form (that is, if gcd(p1+q1, p2+q2)=1) and if q1+q2 <= n. Hence the > second row is [0/1, 1/2, 1/1] and so forth.The q1+q2 rule leads not to a Farey sequence, but to what we've called a "Mann" sequence. For the Farey sequence, the rule is reduced form and q1 <= n and q2 <=n. See Hardy and Wright, for example. Anyhow, the rest looks very interesting . . . what post was it inspired by . . . and how does it relate to tuning?
Message: 852 - Contents - Hide Contents Date: Fri, 24 Aug 2001 07:50:04 Subject: Re: Jacks From: genewardsmith@j... --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> The q1+q2 rule leads not to a Farey sequence, but to what we've > called a "Mann" sequence. For the Farey sequence, the rule is > > reduced form and q1 <= n and q2 <=n. > > See Hardy and Wright, for example.I just saw Hardy and Wright, and they say F_5, the fifth Farey sequence, is just what I said it would be. I defined it interatively, as for instance Niven and Zuckerman do, and Hardy and Wright define it directly, but either way it comes out the same. This is a completely standard definition in elementary number theory, but I'm afraid I don't know what a Mann sequence is--from the way you refer to it, it seems it is not a standard definition.> Anyhow, the rest looks very interesting . . . what post was it > inspired by . . . and how does it relate to tuning?It was inspired by something you posted saying the Blackjack was derived from 36/35 (a high jack) and 81/80, 225/224, and 2401/2400 (jumping jacks.) This suggests how to find something similar; I thought I would write up an explanation for why certain superparticular intervals keep popping up in music theory.
Message: 853 - Contents - Hide Contents Date: Fri, 24 Aug 2001 10:51:47 Subject: Half-octave equivalence (was: Chromatic = commatic?) From: Graham Breed Paul wrote:>> The left hand column defines the generator mapping, which we can > write as >> [ 1] >> -2*[-2] >> [-2] >> >> The minus sign isn't important. The 2 tells us the new interval of >> equivalence is half the old one. >> That's your interpretation. But it isn't correct! We have 10 > equivalence classes in this group, and this half-octave takes you > from one equivalence class to another -- not to the same one.We have 10 notes in the periodicity block, but only 5 equivalence classes. We also have a 24 note periodicity block with only 12 equivalence classes. Or would you prefer to inflate the equivalence interval to 2 octaves in that case? Here, it's simple to redefine the unison vectors to give a 5 note periodicity block with half- octave equivalence: [-1 2 0] [ 0 1 -1] [-2 0 -1] Which means, for my interpretation to be correct, we do need to define "small" for unison vectors.> gives us the mapping by the generator modulo this new interval of>> equivalence. 16:15 is [-1 -1 0] times >> >> [ 1] >> [-2] >> [-2] >> >> or 1 generator. 3:2 is [1 0 0] or 1 generator. Where does the > algebra>> tell us these two intervals are not equivalent? >> Where does it tell us they are? Yes, they're _both_ generators. > Because of the symmetry, there is more than one possible generator. > Think of a prime-numbered ET -- how many generators does that have?Any ET has one generator. Although in octave equivalent terms it has none, because all notes are equivalent. "Octave equivalence" means the octave is considered equivalent to a unison. How can a unison be divided into equal steps? An octave equivalent linear temperament also has one generator. If two different intervals span the same number of generators, they are equivalent in that temperament. That's the case here with the approximations to 3:2 and 16:15. Graham
Message: 855 - Contents - Hide Contents Date: Fri, 24 Aug 2001 19:14:04 Subject: Re: Jacks From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >>> The q1+q2 rule leads not to a Farey sequence, but to what we've >> called a "Mann" sequence. For the Farey sequence, the rule is >> >> reduced form and q1 <= n and q2 <=n. >> >> See Hardy and Wright, for example. >> I just saw Hardy and Wright, and they say F_5, the fifth Farey > sequence, is just what I said it would be.Hmm . . . isn't it 0 1 1 1 2 1 3 2 3 4 1 -, -, -, -, -, -, -, -, -, -, - ? 1 5 4 3 5 2 5 3 4 5 1 This is commonly known as the "Farey series of order 5" -- I thought I remembered seeing it in Hardy and Wright this way -- is it a "series" vs. "sequence" thing? Anyhow, it seems that the "determinant" of two consecutive fractions being equal to one is a property of many differently-defined series -- for example, a product limit on numerator times denominator, or a layer of the Stern-Brocot tree . . .> I defined it interatively, > as for instance Niven and Zuckerman do, and Hardy and Wright define > it directly, but either way it comes out the same.Maybe I misunderstood your iterative definition -- does my rule correspond to Hardy and Wright's direct derivation?> This is a > completely standard definition in elementary number theory, but I'm > afraid I don't know what a Mann sequence is--from the way you refer > to it, it seems it is not a standard definition.No, Mann wrote a book called _Analytic Study of Harmonic Intervals_ or something like that.>>> Anyhow, the rest looks very interesting . . . what post was it >> inspired by . . . and how does it relate to tuning? >> It was inspired by something you posted saying the Blackjack was > derived from 36/35 (a high jack) and 81/80, 225/224, and 2401/2400 > (jumping jacks.)81/80 shouldn't be there.> This suggests how to find something similar; I > thought I would write up an explanation for why certain > superparticular intervals keep popping up in music theory.I'll have to look at it more closely . . . I posted my own explanation for that recently, in a discussion with Monz . . .
Message: 856 - Contents - Hide Contents Date: Fri, 24 Aug 2001 19:19:34 Subject: Re: Half-octave equivalence (was: Chromatic = commatic?) From: Paul Erlich --- In tuning-math@y..., "Graham Breed" <graham@m...> wrote:> Paul wrote: >>>> The left hand column defines the generator mapping, which we can >> write as >>> [ 1] >>> -2*[-2] >>> [-2] >>> >>> The minus sign isn't important. The 2 tells us the new interval > of>>> equivalence is half the old one. >>>> That's your interpretation. But it isn't correct! We have 10 >> equivalence classes in this group, and this half-octave takes you >> from one equivalence class to another -- not to the same one. >> We have 10 notes in the periodicity block, but only 5 equivalence > classes.I'm afraid that's an incorrect interpretation.> We also have a 24 note periodicity block with only 12 > equivalence classes.That's a very different case! It's pathological. We didn't use the generators of the kernel. The decatonic case is not pathological -- the generators of the kernel clearly define 10 equivalence classes, not 5. Gene?> Or would you prefer to inflate the equivalence > interval to 2 octaves in that case? Here, it's simple to redefine > the unison vectors to give a 5 note periodicity block with half- > octave equivalence:That's obvious. But why do that?> > [-1 2 0] > [ 0 1 -1] > [-2 0 -1] > > Which means, for my interpretation to be correct, we do need to > define "small" for unison vectors.I agree that unison vectors should be "small", but I don't think your interpretation (here) is correct!>>> gives us the mapping by the generator modulo this new interval of>>> equivalence. 16:15 is [-1 -1 0] times >>> >>> [ 1] >>> [-2] >>> [-2] >>> >>> or 1 generator. 3:2 is [1 0 0] or 1 generator. Where does the >> algebra>>> tell us these two intervals are not equivalent? >>>> Where does it tell us they are? Yes, they're _both_ generators. >> Because of the symmetry, there is more than one possible generator. >> Think of a prime-numbered ET -- how many generators does that have? >> Any ET has one generator.Incorrect. 7-tET, for example, can be generated by 1/7 octave, 2/7 octave, 3/7 octave . . .> Although in octave equivalent terms it has > none, because all notes are equivalent. Ridiculous. :) > "Octave equivalence" means > the octave is considered equivalent to a unison. How can a unison be > divided into equal steps?By going around in a circle.
Message: 857 - Contents - Hide Contents Date: Fri, 24 Aug 2001 19:19:56 Subject: Re: EDO consistency and accuracy tables (was: A little research...) From: Paul Erlich --- In tuning-math@y..., BobWendell@t... wrote:> Well, thanks to you, I'm leaning toward 72-tET now, since it has the > obvious advantage of greater simplicity and is apparently consistent > through 7-limit. > > Gratefully, > BobIt's consistent through 17-limit!
Message: 858 - Contents - Hide Contents Date: Fri, 24 Aug 2001 19:20:47 Subject: Re: EDO consistency and accuracy tables (was: A little research...) From: Paul Erlich --- In tuning-math@y..., BobWendell@t... wrote:> Well, thanks to you, I'm leaning toward 72-tET nowBut it's useless for adaptive JI. You'll have to deal with drifts or shifts of almost a full comma in typical diatonic triadic progressions.
Message: 860 - Contents - Hide Contents
Date: Fri, 24 Aug 2001 22:00:22
Subject: Consistency, Smith style
From: genewardsmith@j...
I've used a measure which is related to the idea of consistency
proposed here, which I would like to explain.
Let w be an odd number, and let p_d <= w be the largest prime less
than or equal to w, and suppose that there are d primes p_i less than
or equal to l. Let h:G_p_d ->> Z be a homomorphism (more precisely
epimorphism, meaning onto) giving a p_d-limit et. Let {q_i} be the
set of rational numbers q_i >= 1 which are ratios of any two odd
numbers less than or equal to w, and let n = h(2). We define the w-
consistent goodness measure of h as
n^(1/d) * max(abs(n*log_2(q_i) - h(q_i))
In the most interesting cases, the homomorphism h will simply be the
homomorphism h_n obtained by rounding n*log_2(p_i) to the nearest
integer; this lets us define the w-consistent goodness of an integer
n, by setting
cons(w, n) = n^(1/d) * max(abs(n*log_2(q_i)) - h_n(q_i))
The d-th root of n is introduced because it is the appropriate
multiplier according to theorems relating to the simultaneous
Diophantine approximation of d independent numbers.
Here are two tables by way of example:
From 1 to 10000, 5-consistent measure cons(5, n) < 1
1 .736965595
2 .7439736471
3 .4245472985
4 .679700008
5 .8728704449
7 .6706891205
12 .5418300757
15 .8735997285
19 .5083949041
31 .8578063580
34 .6488389972
53 .4527427539
65 .7839449193
118 .4134352529
171 .6499654470
289 .9207676
441 .6622791
559 .9976240155
612 .5676032129
730 .6113208564
1171 .7597497149
1783 .5008376597
2513 .5396476355
4296 .2748910262
6809 .7979361504
8592 .7776946207
From 1 to 10000, 9-consistent measure cons(9, n) < 1.25
1 1.152003094
2 .9136193505
4 1.078956495
5 .7303891055
7 1.151607929
10 1.120372697
12 .8032252875
19 .9065721690
22 .9640922367
27 1.236074910
31 .9044163694
41 .8004237371
46 1.181094402
53 1.023130352
72 .9759757458
99 .8207791216
130 1.213753821
171 .3202177291
270 .7772103754
342 .8068974155
441 .8113651760
612 .9093921787
935 1.231274556
1106 1.219566982
1277 1.192780688
1848 1.206088177
2954 1.055468357
3125 .6018359509
3296 1.116123065
3566 1.147361792
6691 .9930572626
8539 1.219825812
Message: 861 - Contents - Hide Contents Date: Fri, 24 Aug 2001 22:02:52 Subject: Re: Consistency, Smith style From: genewardsmith@j... --- In tuning-math@y..., genewardsmith@j... wrote:> Let w be an odd number, and let p_d <= w be the largest prime less > than or equal to w, and suppose that there are d primes p_i less than > or equal to l.Sorry, an ascii "l" looks like the number "1", so I changed the l's to w's but missed this one. It should be p_i less than or equal to w.
Message: 862 - Contents - Hide Contents Date: Fri, 24 Aug 2001 23:17:15 Subject: Re: EDO consistency and accuracy tables (was: A little research...) From: Paul Erlich --- In tuning-math@y..., BobWendell@t... wrote:> Well, rats! LOL....Maybe you have some other recommendation that > would satisfy the stated goals and still be useful for adaptvie JI?Are cents really so bad?> Or maybe for compositional purposes, I don't need to be concerned > with that?If you're not planning on worrying about abrupt commatic shifts in your melodies (that is, if they don't bother you at all), then maybe 111-tET would be fine for you. Personally, I see 152-tET as my "Universal tuning". But I'm not going past 11-limit, and I care very much about certain melodic systems.
Message: 863 - Contents - Hide Contents Date: Fri, 24 Aug 2001 23:19:00 Subject: Re: Consistency, Smith style From: Paul Erlich I'll have to look at this later, Gene. Sounds very interesting and not unlike some things I've mentioned on the tuning list about 5 years ago when it was on the Mills server.
Message: 864 - Contents - Hide Contents Date: Fri, 24 Aug 2001 23:58:44 Subject: Re: Consistency, Smith style From: genewardsmith@j... --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> I'll have to look at this later, Gene. Sounds very interesting and > not unlike some things I've mentioned on the tuning list about 5 > years ago when it was on the Mills server.Incidentally, if you or anyone else is interested in Maple routines for functions such as cons(w, n) I could post them.
Message: 865 - Contents - Hide Contents Date: Sat, 25 Aug 2001 00:07:15 Subject: Re: Jacks From: genewardsmith@j... --- In tuning-math@y..., genewardsmith@j... wrote:> Let me start by reviewing Farey sequences. The first row of the Farey > sequence is [0/1, 1/1] and the nth row is obtained by inserting the > fraction (p1+q1)/(p2+q2) between p1/q1 and p2/q2 if it is in reduced > form (that is, if gcd(p1+q1, p2+q2)=1) and if q1+q2 <= n. Hence the > second row is [0/1, 1/2, 1/1] and so forth.This should have been (p1+p2)/(q1+q2) between p1/q1 and p2/q2, of course. Was that the problem?
Message: 866 - Contents - Hide Contents Date: Sat, 25 Aug 2001 00:14:16 Subject: Re: Now I think "the hypothesis" is true :) From: genewardsmith@j... --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> In my proof, I actually get at _two different_ (but mathematically > equivalent) definitions of MOS. One concerns not a "two-scale-step" > condition, as you say, but rather a "Myhill" condition, which says > that every generic interval, not just steps but _any_ interval, aside > from the interval of repetition, will come in exactly two step sizes.Dave told me I was barking up the wrong tree with the semiconvergent business, though it seems to me it should imply Myhill's property. Did what I said then strike you as using a wrong definition?
Message: 867 - Contents - Hide Contents Date: Sat, 25 Aug 2001 00:37:48 Subject: Re: Now I think "the hypothesis" is true :) From: Dave Keenan --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >>> In my proof, I actually get at _two different_ (but mathematically >> equivalent) definitions of MOS. One concerns not a "two-scale-step" >> condition, as you say, but rather a "Myhill" condition, which says >> that every generic interval, not just steps but _any_ interval, > aside>> from the interval of repetition, will come in exactly two step > sizes. >> Dave told me I was barking up the wrong tree with the semiconvergent > business, though it seems to me it should imply Myhill's property.Yes it certainly does. We already knew that.> Did what I said then strike you as using a wrong definition?No. The definition is fine. But the conjecture/hypothesis we're trying to prove is tempered-PB = semiconvergent This is where we could really use your help Gene. -- Dave Keenan
Message: 869 - Contents - Hide Contents Date: Sun, 26 Aug 2001 02:38:50 Subject: Re: EDO consistency and accuracy tables (was: A little research...) From: genewardsmith@j... --- In tuning-math@y..., BobWendell@t... wrote:> Comma drifts, if writing original compositions as if one had never > been exposed to any other system, might actually be viewed as > microtonal modulations, and one can write in such a way as to return > home just as we do in 12-tET. We come back home if we chose to > because we compose with the intention of doing so within the > constraints the tuning offers us.A "comma drift" is simply another word for a microtonal modulation, and the idea that avoiding the use of just intonatation will spare us from comma drifts is incorrect. It will set any microtonal modulation which belongs to the kernel to unison, but not any that do not--in fact, these may become exaggerated from what they would have been had just intonation been employed.> If we freely exploit the character of a high-order EDO, why not > conceive things in terms of its indigenous character and modulate > away from and back to a pitch center as that system's character > constrains our freedom to choose as any system does. Why shouldn't > our art reflect the intonational medium in which we are working > rather than superimposed ideas from media alien to it?I think microtonalists should use whatever scale suits them; however it would be well if they understood the structure of the system they intend to use.
Message: 870 - Contents - Hide Contents Date: Sun, 26 Aug 2001 03:47:15 Subject: Re: Now I think "the hypothesis" is true :) From: genewardsmith@j... --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:> No. The definition is fine. > > But the conjecture/hypothesis we're trying to prove is > > tempered-PB = semiconvergent > > This is where we could really use your help Gene.I need to see how "tempered-PB" leads to a tuning. For instance, what does and does not count as a mean-tone tuning? If we take 25/24 and 81/80 as our canonical example, leading to the 7- note scales related to the h_7(2^a*3^b*5^c) = 7*a+11*b+16*c 7-et homomorphism, we can temper out 81/80 (commatic unison) and not 25/24. We then arrive at the mean-tone systems in the 5-limit. We can generate our scale with any number of scale steps relatively prime to 7; for instance 4 steps representing a fifth. If we tune the fifth by setting it to x cents, then x/1200 is log_2 of the approximation to the fifth are using. This will have 4/7 (4 scale steps out of 7) as a semiconvergent if |x/1200 - 4/7| < 1/49, but if it is very far out of this range we are in trouble. In other words, in terms of cents we want x to be in the interval |x - 685.7 ...| < 171.4...; this includes a wide range of tunings for a fifth, but not all tunings whatever.
Message: 871 - Contents - Hide Contents Date: Sun, 26 Aug 2001 03:54:29 Subject: Re: Now I think "the hypothesis" is true :) From: genewardsmith@j... --- In tuning-math@y..., genewardsmith@j... wrote:> this range we are in trouble. In other words, in terms of cents we > want x to be in the interval |x - 685.7 ...| < 171.4...; this > includes a wide range of tunings for a fifth, but not all tunings > whatever.This should be |x - 685.7| < 24.5, sorry.
Message: 872 - Contents - Hide Contents Date: Sun, 26 Aug 2001 20:56:25 Subject: Re: EDO consistency and accuracy tables (was: A little research...) From: Paul Erlich --- In tuning-math@y..., BobWendell@t... wrote:> Hi, Paul! Thanks... > > I've been pondering this a bit, and I'm wondering: > > If I'm only interested in choosing some n-tET as a tool strictly to > to serve as a kind of simplified compositional calculus, do I really > need to be concerned about more than accurate approximation of just > intervals and consistency?Well, sure -- each ET will imply a different set of progressions that drift, and a different set that don't.> > Why ask? Each tuning strategy has its own characteristics and it > occurs to me that a composer needn't and perhaps even shouldn't think > in terms of how it interfaces with the character of others.Sure! But you need to be aware of what those characteristics are!> > Comma drifts, if writing original compositions as if one had never > been exposed to any other system, might actually be viewed as > microtonal modulations, and one can write in such a way as to return > home just as we do in 12-tET. We come back home if we chose to > because we compose with the intention of doing so within the > constraints the tuning offers us. > > If we freely exploit the character of a high-order EDO, why not > conceive things in terms of its indigenous character and modulate > away from and back to a pitch center as that system's character > constrains our freedom to choose as any system does. Why shouldn't > our art reflect the intonational medium in which we are working > rather than superimposed ideas from media alien to it?I agree completely -- but these characteristics exert a powerful effect on one's compositional possibilities. Hence, it seems premature to settle for a given ET as one's "tool" without being sure that one wants to be constrained by its particular behaviors.
Message: 873 - Contents - Hide Contents Date: Sun, 26 Aug 2001 20:58:35 Subject: Re: Now I think "the hypothesis" is true :) From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote: >>> No. The definition is fine. >> >> But the conjecture/hypothesis we're trying to prove is >> >> tempered-PB = semiconvergent >> >> This is where we could really use your help Gene. >> I need to see how "tempered-PB" leads to a tuning. For instance, what > does and does not count as a mean-tone tuning? > > If we take 25/24 and 81/80 as our canonical example, leading to the 7- > note scales related to the h_7(2^a*3^b*5^c) = 7*a+11*b+16*c 7-et > homomorphism, we can temper out 81/80 (commatic unison) and not 25/24. > We then arrive at the mean-tone systems in the 5-limit. We can > generate our scale with any number of scale steps relatively prime to > 7; for instance 4 steps representing a fifth. If we tune the fifth by > setting it to x cents, then x/1200 is log_2 of the approximation to > the fifth are using. This will have 4/7 (4 scale steps out of 7) as a > semiconvergent if |x/1200 - 4/7| < 1/49, but if it is very far out of > this range we are in trouble. In other words, in terms of cents we > want x to be in the interval |x - 685.7 ...| < 171.4...; this > includes a wide range of tunings for a fifth, but not all tunings > whatever.What if we first stick with the case where the chromatic unison vector is unchanged in size -- so in this case, 2/7-comma meantone.
Message: 874 - Contents - Hide Contents
Date: Sun, 26 Aug 2001 21:34:39
Subject: 19-consistent goodness of measurement systems
From: genewardsmith@j...
Here is a list of cons(19, n) for n in the range 1 to 10000. The
function "cons" was calclulated via the following Maple routine:
with(padic,ordp):
cons := proc(w,n)
local i,j,d,e,t,f,h,p,q,u,v;
d := 0;
for i by 2 to w do if isprime(i) then d := d+1; p[d] := i fi od;
e := 0;
for i by 2 to w-2 do
for j from i+2 by 2 to w do
if gcd(i,j) = 1 then e := e+1; q[e] := j/i fi
od
od;
t := ln(2);
for i to d do h[i] := round(evalf(n*ln(p[i])/t)) od;
for i to e do
u[i] := 0; for j to d do u[i] := u[i]+ordp(q[i],p[j])*h[j] od
od;
for i to e do v[i] := abs(evalf(n*ln(q[i])/t-u[i])) od;
f := 0;
for i to e do f := max(f,v[i]) od;
evalf(n^(1/d)*f)
end:
Some idea of the advantages of programming in Maple or similar
computer algebra packages (e.g. Mathematica, Macsyma, Axiom, etc.)
can be discerned from this--the program uses built-in functions to
compute the GCD, determine if an integer is prime, and measure (with
the ordp function) the divisibility of a rational number by a given
prime.
19-consistent measure with cons(19, n) < 1 for n from 1 to 10000
2 .9728631558
7 .9718578174
31 .9504221384
50 .9174777262
80 .9050078510
94 .8876782729
111 .9407450518
121 .8711227952
311 .8268720520
320 .8842773382
364 .9500455011
400 .9876634122
422 .9828907615
436 .9633235237
460 .9202207116
581 .8608767534
742 .9205311588
1178 .8708834628
1578 .8898715197
2000 .8594235879
2460 .9155568082
3395 .9139203767
8539 .8797434553
Nothing on this list really jumps out, though clearly 311 is a very
strong contender even at the 19-limit. Aside from the goodness (or
perhaps it should be badness, since smaller numbers are better)
measure, we might want to note divisibility properties. The 2460
system system is divisible by 12, and 111 and 1578 are at least
divisible by 3, which accomplishes more or less the same thing;
including some systems which have been mentioned but are not on this
list we have:
72 = 6 * 12
111 = 9.25 * 12
612 = 51 * 12
1200 = 100 * 12
1578 = 131.5 * 12
2460 = 205 * 12
If representing other divisions would be useful, we also have:
2460 = 164 * 15 = 60 * 41 = 205 * 12
152 = 8 * 19
171 = 9 * 19
1178 = 62 * 19 = 38 * 31
121 = 5.5 * 22
460 = 10 * 46
742 = 14 * 53
If you count 50 as interesting, you can even add:
80 = 1.6*50, 320 = 6.4*50, 400 = 8*50, 460 = 9.2*50, 1200 = 24*50,
2000 = 40*50, 2460 = 49.2*50, and 3395 = 67.9*50.
Alas, 311 and 8539 are prime!
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